We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair is dense in the space of quantum strategies obtained by using states on the maximal tensor product. We exhibit a number of examples of such pairs that are “nontrivial” in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchberg’s QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP forms an elementary class (in the sense of model theory), but that this class does not admit an effective axiomatization.
We introduce the notion of a Tsirelson pair of C*-algebras, which is a pair of C*-algebras for which the space of quantum strategies obtained by using states on the minimal tensor product of the pair is dense in the space of quantum strategies obtained by using states on the maximal tensor product. We exhibit a number of examples of such pairs that are “nontrivial” in the sense that the minimal tensor product and the maximal tensor product of the pair are not isomorphic. For example, we prove that any pair containing a C*-algebra with Kirchberg’s QWEP property is a Tsirelson pair. We then introduce the notion of a C*-algebra with the Tsirelson property (TP) and establish a number of closure properties for this class. We also show that the class of C*-algebras with the TP forms an elementary class (in the sense of model theory), but that this class does not admit an effective axiomatization.
Tsirelson once claimed that the set of quantum correlations, defined by strategies of non-local two-player games, does not depend on which of two possible models is chosen: the tensor product model or the commuting operator model. He later came back from this claim, and the resulting conjecture is now known as Tsirelson’s problem. The problem has since been proven equivalent to notoriously hard problems in operator theory, such as the Connes’ Embedding Problem and the QWEP conjecture. In this master thesis, we look at the finite dimensional case of Tsirelson’s problem, working out all the details of an existing proof and giving a new, shorter proof which also extends to the nuclear case. Moreover, we give an overview of the equivalence of Tsirelson’s problem and two of Kirchberg’s conjectures, including the QWEP conjecture. Finally, we give some results and considerations for the three-player case of Tsirelson’s problem. The appendix contains proofs of many related results used throughout the thesis, and also a beginner’s introduction to quantum mechanics.
Entanglement in C∗-algebras: tensor products of state spaces
We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C∗-algebras. Relatedly, we study entanglement in (infinite dimensional) C∗-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C∗-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C∗-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C∗-algebras. We show that the minimal and maximal tensor products of state spaces of C∗-algebras agree precisely when one of the two C∗-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C∗-algebras.
Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C∗-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C∗-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C∗-algebras.
In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the algebraic tensor product of p-adic Hilbert spaces. We next define a suitable norm on this linear space. It turns out that, in the p-adic framework, this norm is the analogue of the projective norm associated with the tensor product of real or complex normed spaces. Eventually, by metrically completing the resulting p-adic normed space, and equipping it with a suitable inner product, we obtain the tensor product of p-adic Hilbert spaces. That this is indeed the correct p-adic counterpart of the tensor product of complex Hilbert spaces is also certified by establishing a natural isomorphism between this p-adic Hilbert space and the corresponding Hilbert–Schmidt class. Since the notion of subspace of a p-adic Hilbert space is highly nontrivial, we finally study the tensor product of subspaces, stressing both the analogies and the significant differences with respect to the standard complex case. These findings should provide us with the mathematical foundations necessary to explore quantum entanglement in the p-adic setting, with potential applications in the emerging field of p-adic quantum information theory.
Starting from the meaning of the conjugate of a complex Hilbert space, including a related application of the theorem of Fréchet-Riesz (by which an analysis of semilinear operators can be reduced to - linear - operator theory) to a revisit of applications of nuclear and absolutely p-summing operators in algebraic quantum field theory in the sense of Araki, Haag and Kastler (p=2) and more recently in the framework of general probabilistic spaces (p=1), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in algebraic quantum field theory. In doing so, we establish a canonical isometric isomorphism between the Hilbert spaces H⊗2(K⊗2L) and (H⊗2K)⊗2L (Theorem 3.8) and revisit the role of trace class operators. A few applications are specified, including the appropriateness of the class of Hilbert-Schmidt operators and an implied Banach operator ideal representation of the tensor product of two complex Hilbert spaces H⊗2K (Proposition 3.4) and a purely linear algebraic description of the quantum teleportation process (Example 3.10).
Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.
We show that the negation of the Tsirelson’s conjecture can be understood as, and in fact follows from, the ZFC - genericity of infinite sequences of the QM outcomes. We extend the result of Landsman that there holds the equivalence of the statistics of the entire sequences with the statistic of the single QM measurements in such sequences, over stratified infinite tensor products of the finite dimensional spaces. The stratification is based on uncomputable Turing classes and also on the Solovay genericity and the equivalence is now corrected by the appearance of certain self-adjoint operator. From studies on algorithmic randomness follows that these two properties are in a sense orthogonal one to the other for random sequences. We show two models of ZFC which indeed separate them. This is based on the classic results for Cohen and random forcings in set theory and also on Takeuti’s early results on Boolean-valued models and quantum mechanics. This separation is the tool for showing that not all correlations between commuting operators on infinite dimensional Hilbert spaces can be reproduced on a tensor product of two spaces with the corresponding self-adjoint operators. We discuss the findings also from the perspective of eventual practical applications. The supplementary subsections contain discussion of QM with the set theory component along with Solovay randomness of the outcomes.